Wedderburn-Mal'tsev theorem

Let $ A $ be a finite-dimensional associative algebra (cf. Associative rings and algebras) over a field $ F $ with radical $ N $, and let the quotient algebra $ A/N $ be a separable algebra (for algebras over a field of characteristic zero this is always true). Then $ A $ can be decomposed (as a linear space) into a direct sum of the radical $ N $ and some semi-simple subalgebra $ S $:

$$ A = N \oplus S, $$

and if there exists another decomposition $ A = N \oplus {S _ {1} } $, where $ S _ {1} $ is a semi-simple subalgebra, then there exists an automorphism $ \phi $ of the algebra $ A $ which maps $ S $ onto $ S _ {1} $( the automorphism $ \phi $ is inner, i.e. there exist elements $ a, a ^ \prime \in A $ such that $ a \cdot a ^ \prime = a ^ \prime \cdot a = 0 $ and $ x \phi = a \cdot x \cdot a ^ \prime $ for all $ x \in A $, where $ x \cdot y = x + y + xy $). The existence of this decomposition was shown by J.H.M. Wedderburn [1] and the uniqueness, up to an automorphism of the semi-simple term, was proved by A.I. Mal'tsev [2]. This theorem, together with Wedderburn's theorem (cf. Associative rings and algebras) on the structure of semi-simple algebras constitutes the central part of the classical theory of finite-dimensional algebras.

References

[1]	J.H.M. Wedderburn, "On hypercomplex numbers" Proc. London Math. Soc. (2) , 6 (1908) pp. 77–118
[2]	A.I. Mal'tsev, "On the representation of an algebra as a direct sum of the radical and a semi-simple subalgebra" Dokl. Akad. Nauk SSSR , 36 : 1 (1942) pp. 42–45 (In Russian)
[3]	A.A. Albert, "Structure of algebras" , Amer. Math. Soc. (1939)
[4]	C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962)

Comments

A similar theorem holds for Lie algebras. Let $ \mathfrak g $ be a finite-dimensional Lie algebra over a field of characteristic zero with radical $ \mathfrak r $. Then there exists a semi-simple subalgebra $ \mathfrak h $ of $ \mathfrak g $ such that $ \mathfrak g = \mathfrak h \oplus \mathfrak r $. Such a decomposition is called a Levi decomposition and $ \mathfrak h $ is called a Levi factor or Levi subalgebra. It is unique up to inner automorphisms.

References

[a1]	N. Jacobson, "Lie algebras" , Dover, reprint (1962) pp. 91ff ((also: Dover, reprint, 1979))